Circulating type planetary roller screw structure optimization design method

Abstract

The invention discloses an optimal design method for a circulating type planetary roller screw structure. The method comprises the following steps: firstly, establishing a multi-body contact mathematical model of a circulating type planetary roller screw; constructing a Bayesian learning model based on variational inference in combination with an active learning Kriging strategy; constructing a multi-objective optimization model, and determining design structure parameters, optimization objectives and constraint conditions; and solving the multi-objective optimization model by combining a Bayesian learning model and an NSGA-III algorithm, so as to obtain an optimal parameter combination when the contact points among the structures of the circulating planetary roller screw are constrained in a given range and the contact stress is minimized at the same time. According to the invention, the contact stress can be effectively reduced, the contact fatigue can be delayed, the transmission reliability can be improved, and the requirements of high-precision and high-load application can be met.

Description

Technical Field

[0001] The present invention relates to the technical field of lead screw structure design, and particularly relates to an optimized design method for a cyclic planetary roller screw structure. Background Art

[0002] A cyclic planetary roller screw (RPRSM) mainly consists of a threaded screw, a nut, rollers, and a pair of cam rings. Its rollers adopt an equidistant groove structure, eliminating the helix angle of the traditional thread and significantly reducing the processing cost. Multiple rollers are evenly distributed on the circumference of the screw by a cage. The length of the cage is slightly longer than that of the rollers to allow the rollers to move axially within the nut. Due to the relative axial movement of the rollers within the nut, the cyclic reset of the rollers needs to be achieved through the bumps on the cam rings. The nut is designed with a non-threaded area. When the rollers enter this area, they disengage from the screw, thus completing the reset process. The cyclic planetary roller screw features a small lead, high resolution, and high positioning accuracy, and at the same time has a high load capacity and axial rigidity, capable of providing ultra-precise driving performance. Therefore, it is widely used in fields such as optical instruments, valve control technology, and hexapod robot drives. Although there are already mature commercial products available, the comprehensive and systematic theoretical research on cyclic planetary roller screws is still relatively limited.

[0003] The cyclic planetary roller screw involves complex multi-body contact interactions because the rollers not only contact the screw and the nut but also the cam rings. This multi-body contact characteristic makes the distribution of its contact stress complex, easily leading to problems such as local stress concentration, increased wear, and fatigue failure.

[0004] Existing design methods all adopt mathematical modeling for overall design simulation and then fine-tune some parameters based on experience. This method highly depends on the experience of the designer, and during actual use, the vibration, noise, and running accuracy generated cannot be systematically controlled, resulting in low overall structural stability and affecting the wide application of cyclic planetary roller screws. Moreover, the non-linear characteristics in contact performance calculation significantly increase the calculation cost, which is particularly prominent in optimization problems. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide an optimized design method for a cyclic planetary roller screw structure, which can effectively reduce contact stress, delay contact fatigue, and improve the reliability of transmission, meeting the requirements of high-precision and high-load applications.

[0006] To solve the above technical problem, the present invention provides an optimized design method for a cyclic planetary roller screw structure, including the following steps:

[0007] Establish a mathematical model of multi-body contact of the cyclic planetary roller screw;

[0008] Construct a multi-objective optimization model based on the mathematical model of multi-body contact, and determine the design structure parameters, optimization objectives, and constraint conditions; construct a Bayesian learning model based on variational inference combined with the active learning Kriging strategy;

[0009] By combining the Bayesian learning model with the NSGA-III algorithm, solve the multi-objective optimization model to obtain the optimal parameter combination when the contact points between the structures of the circulating planetary roller screw are constrained within a given range and the contact stress is minimized simultaneously.

[0010] Furthermore, the external thread profile of the screw in the circulating planetary roller screw is triangular, the internal thread profile of the nut of the circulating planetary roller screw is the same as that of the screw and has a threadless area, the roller of the circulating planetary roller screw is a non-threaded structure and is an equidistant groove-like structure and the groove pitch of the roller is equal to the lead of the screw, a pair of cam rings of the circulating planetary roller screw are installed at both ends of the nut and coincide with the axis of the nut, convex blocks are provided on the cam rings, the height of the convex blocks is equal to the lead of the screw and the inclination angle of the convex blocks is equal to the flank angle of the roller.

[0011] Furthermore, the mathematical model of multi-body contact includes:

[0012] The parametric equation of the tooth profile of the screw to determine the unit normal vector of any point on the screw thread tooth;

[0013] The parametric equation of the tooth profile of the nut to determine the unit normal vector of any point on the nut thread tooth;

[0014] The parametric equation of the tooth profile of the roller to determine the unit normal vector of any point on the roller groove surface;

[0015] The parametric equation of the inclined plane of the convex block to determine the parameter value range of the points on the inclined plane of the convex block.

[0016] Furthermore, determine the contact positions and contact stresses between the roller and the screw, between the roller and the nut, and between the roller and the convex block.

[0017] Furthermore, the design structure parameters include the scaling factor of the roller tooth profile radius, the flank angle of the roller, the flank angle of the screw, and the flank angle of the nut, where the inclination angle of the convex block of the cam ring is equal to the flank angle of the roller;

[0018] The optimization objectives include the contact stress between the roller and the screw, the contact stress between the roller and the nut, and the contact stress between the roller and the convex block;

[0019] The constraint objectives include tooth profile constraint, axis clearance constraint, and contact point constraint. The tooth profile constraint is used to limit the tooth tip thickness and tooth root thickness. The axis clearance constraint is used to prevent interference between the roller and the lead screw and between the roller and the nut. The contact point constraint is used to accurately limit the contact point position.

[0020] Furthermore, the multi-objective optimization model is as follows after setting the constraint conditions:

[0021] find x = [k E , k βN , k βR , k βS T

[0022] min σ(x, μ) = [σ RN , σ RS , σ RP T

[0023]

[0024] where k E is the scaling factor of the roller tooth profile radius, k βN , k βR and k βS are the scaling factors of the flank angles of the nut, roller, and lead screw respectively. x is a vector, μ is a design constant, and σ(x, μ) is the final objective function;

[0025] h cN , h cR , h cS represent the tooth tip thicknesses of the nut, roller, and lead screw respectively, and h aN , h aR , h aS represent the tooth root thicknesses of the nut, roller, and lead screw respectively; P N and P S represent the pitches of the nut and lead screw respectively; P R is the slot pitch of the roller; and represent the axial clearances between the roller and the corresponding contact surfaces of the lead screw and nut respectively. γ RS and γ S represent the half-angles of the overlapping regions between the roller groove surface and the lead screw thread surface respectively; γ RN and γ N represent the half-angles of the overlapping regions between the roller groove surface and the nut thread surface respectively; r S0 is the nominal radius of the lead screw, r N0 is the nominal radius of the nut, Δ RS is the bandwidth of the contact point position fluctuation range between the roller and the lead screw, and Δ RN ​​is the bandwidth of the fluctuation range of the contact point position between the roller and the nut; and respectively represent the contact deflection angles of the roller and the nut, and respectively represent the contact radii of the roller and the nut; and respectively represent the contact deflection angles of the roller and the lead screw, and respectively represent the contact radii of the roller and the lead screw.

[0026] Furthermore, the tooth tip thickness and the tooth root thickness meet the following requirements:

[0027]

[0028] The axial clearance between the corresponding contact surfaces of the roller and the lead screw and the nut meets the following requirements:

[0029]

[0030] The half angle of the overlapping area between the roller and the lead screw meets the following requirements:

[0031]

[0032] The half angle of the overlapping area between the roller and the nut interface meets the following requirements:

[0033]

[0034] Furthermore, the sampling strategy of variational inference combined with active learning Kriging is as follows:

[0035] Step 1: Use the Latin hypercube sampling method to conduct preliminary sampling in a small-scale sample space and establish a rough initial Kriging model;

[0036] Step 2: Use the initial Kriging model as the initial estimate of the posterior distribution p(f|D), where f represents the objective function and D represents the observed data;

[0037] Step 3: Select a new variational distribution q(f; λ) to approximate the true posterior distribution p(f|D), where λ is the variational parameter;

[0038] Step 4: Calculate the evidence lower bound;

[0039] Step 5: Find the optimal variational parameter λ* by maximizing the evidence lower bound;

[0040] Step 6: Use the optimized posterior distribution q(f; λ*) to generate a large number of candidate samples;

[0041] Step 7: Use the density-based spatial clustering method to divide the candidate samples into n different and parallel active regions;

[0042] Step 8: Generate parallel training points in the n active regions These training points are added to the optimization process as candidate points, and the candidate point with the maximum expected improvement is selected as the new sampling point for the next iteration within each active region;

[0043] Step 9: Check the ratio between the maximum expected improvement of all samples in the current iteration and the minimum value among the current predicted values to determine whether the given convergence threshold is met. If it is met, end the process; otherwise, add the new sampling point as an additional training point to the training point set, update the adaptive region, and update the Kriging model, then return to Step 2 to continue execution.

[0044] Furthermore, use the Bayesian learning model as the surrogate model for the optimization objective in the multi-objective optimization model, and the NSGA-III algorithm uses the surrogate model to evaluate the objective function.

[0045] Advantages of the present invention:

[0046] Through the structural optimization design of the circulating planetary roller screw, this patent aims to minimize the contact stress, thereby reducing frictional losses and improving the reliability and service life of the circulating planetary roller screw. Optimizing the contact stress can not only improve the load distribution, reduce vibration and noise, but also enhance the smoothness and positioning accuracy of the transmission, meeting the requirements of high-precision and high-load applications. In addition, the optimization design promotes the development of multi-body contact mechanics, optimization algorithms, and advanced manufacturing technologies, providing important theoretical support and technical guarantee for the design and application of high-performance transmission systems.

[0047] The combination of variational inference (VI) and active learning Kriging (ALK) can reduce the computational complexity by optimizing the approximate posterior distribution, making it more suitable for large-scale data analysis. At the same time, through the adaptive sampling strategy, high-information samples are preferentially selected, reducing redundant calculations while improving the accuracy of the surrogate model. Thus, an efficient Bayesian learning model is obtained, which can improve the analysis accuracy and optimization efficiency, enhance the computational efficiency, and also strengthen the robustness of the optimization process. Description of the drawings

[0048] Figure 1 is the structural diagram of the circulating planetary roller screw of the present invention;

[0049] Figure 2 is the schematic diagram of the contact between the roller and the convex block of the present invention;

[0050] Figure 3 is the schematic diagram of the meshing between the roller groove surface and the screw and nut thread surfaces of the present invention;

[0051] Figure 4 It is a schematic diagram of the engagement between the roller groove surface and the convex block inclined surface of the present invention;

[0052] Figure 5 It is a force analysis diagram of the roller, the lead screw, and the nut of the present invention;

[0053] Figure 6 It is a force analysis diagram of the roller and the convex block of the present invention;

[0054] Figure 7 It is a VI-ALK strategy flowchart of the present invention;

[0055] Figure 8 It is a diagram of the optimization process of the test function based on the Bayesian learning model of the present invention;

[0056] Figure 9 It is a diagram of the constraint conditions for the structural optimization of the cyclic planetary roller screw of the present invention;

[0057] Figure 10 It is the optimization history of the design variables of the present invention and the corresponding 3D scatter plot;

[0058] Figure 11 It is the optimization history and distribution diagram of the objective function of the present invention;

[0059] Figure 12 It is a comparison diagram before and after the optimization of the local contact characteristics between the roller and the lead screw of the present invention;

[0060] Figure 13 It is a comparison diagram before and after the optimization of the local contact characteristics between the roller and the nut of the present invention;

[0061] Figure 14 It is a comparison diagram before and after the optimization of the local contact characteristics between the roller and the convex block of the present invention. Detailed implementation manners

[0062] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited do not limit the present invention.

[0063] Refer to Figure 1 As shown, an embodiment of the structural optimization design method of the cyclic planetary roller screw of the present invention, the cyclic planetary roller screw structure includes a lead screw 1 and a nut 2, a plurality of rollers 3 are arranged between the lead screw and the nut, the installation positions of the plurality of rollers are determined by a cage 4, the rollers are engaged with the threads on the lead screw and the nut, cam rings 5 are arranged at both ends of the cage, a thread-free area 6 is arranged on the inner wall of the nut, and a convex block 7 is arranged on one side surface of the cam ring located on the rollers for realizing the cyclic reset of the rollers.

[0064] Based on the above structure, the proposed structural optimization design method of the Recirculating Planetary Roller Screw Mechanism (RPRSM) includes first establishing a mathematical model of the multi-body contact of the RPRSM; then constructing a Bayesian learning model based on variational inference combined with the active learning Kriging strategy; combining the Bayesian learning model with the NSGA-III algorithm to form a multi-objective optimization model, and determining the design structure parameters, optimization objectives, and constraint conditions; finally, solving through the multi-objective optimization model to obtain the optimal parameter combination when the contact stresses between the structures of the RPRSM reach the minimum simultaneously. This application constructs a Bayesian learning model for the RPRSM. Bayesian learning, as a probabilistic machine learning method, provides a powerful theoretical framework for uncertainty quantification and adaptive model updating. By combining variational inference (VI), Bayesian learning can efficiently approximate the intractable posterior distribution in complex high-dimensional nonlinear problems, significantly improving the computational efficiency. At the same time, Bayesian learning and the active learning Kriging (ALK) method complement each other: Bayesian learning focuses on high-uncertainty regions by guiding sample selection, providing strong support for ALK, thus significantly improving the accuracy of the surrogate model while reducing the computational cost. Based on this advantage, a Bayesian learning model based on the VI-ALK strategy proposed in this application constructs a high-precision RPRSM surrogate model by organically combining VI and ALK, not only improving the computational efficiency but also enhancing the robustness of the optimization process.

[0065] For establishing the mathematical model of the RPRSM, its meshing characteristics, mechanical behavior, and kinematic characteristics can be analyzed in detail, and the modeling is as follows:

[0066] 1. Screw: The external thread profile of the screw is triangular, and the parametric equation of its tooth profile is:

[0067] where P S , h S , β S , r S0 , r S1 and r S2 are respectively the pitch, tooth thickness, tooth flank angle, nominal radius, outer diameter radius, and inner diameter radius of the screw, r S is the length parameter of the polar coordinate of a point on the screw rod, r S1 and r S2 are used to constrain the range of r S , r S2 ≤r S ≤r S1 . The number of thread starts of the screw is n, and its lead is l S = nP S。The unit normal vector at any point on the screw thread tooth of the lead screw is:

[0068] In the formula α S is the angular parameter of the polar coordinates of a point on the lead screw. (r S , α S ) are the polar coordinate parameters of a point on the lead screw.

[0069] 2. Nut: The inner thread tooth profile of the nut is the same as that of the lead screw, but there is a non-threaded area. The parametric equation of the tooth profile in the threaded area is:

[0070] In the formula, P N , h N , β N , r N0 , r N1 and r N2 are respectively the pitch, tooth thickness, flank angle, nominal radius, outer diameter radius and inner diameter radius of the nut. r N is the length parameter of the polar coordinates of a point on the lead screw. r N1 and r N2 are used to constrain the range of r N , r N2 ≤ r N ≤ r N1 .

[0071] The unit normal vector at any point on the nut thread tooth is:

[0072] In the formula α N is the angular parameter of the polar coordinates of a point on the lead screw. (r N , α N ) are the polar coordinate parameters of a point on the lead screw.

[0073] 3. Roller: The roller is a non-threaded structure and is an equidistant grooved structure. The groove pitch of the roller is equal to the lead of the lead screw. The parametric equation of its tooth profile is:

[0074] In the formula, P R , h R , β R , r R0 , r R1 and r R2 are respectively the groove pitch, tooth thickness, flank angle, nominal radius, outer diameter radius and inner diameter radius of the roller. r R is the length parameter of the polar coordinates of a point on the lead screw. r R1 and r R2 are used to constrain the range of r R , rR2 ≤ r R ≤ r R1 。

[0075] The tooth profile of the roller is usually designed as an arc to ensure point contact. When the center of the arc profile is on the axis of the roller, its radius is r E0 , and the calculation method is as follows:

[0076] Wherein, an additional scaling factor k is introduced E to adjust the tooth profile radius of the roller. Therefore, the actual radius is r E = k E r E0 . The unit normal vector at any point on the roller groove surface is:

[0077] In the formula: α R is the angular parameter of the polar coordinates of a point on the lead screw. (r R , α R ) are the polar coordinate parameters of a point on the lead screw.

[0078] 4. Cam ring: A pair of cam rings are installed at both ends of the nut and coincide with the axis of the nut. The convex blocks are located on both sides of the threadless area of the nut. In addition, the height of the convex block on the cam ring is equal to the lead of the lead screw, and the inclination angle β of the convex block P is equal to the flank angle β of the roller R . Referring to Figure 2 shown, the parametric equation of the convex block inclined plane is: Φ(r P , α P ) = [r P cosα P , r P sinα P , r P tanβ P (sinγ P - sinα P )]; In the formula, where γ P is half of the included angle between the two inclined planes of the convex block, r P1 and r P2 respectively represent the inner radius and outer radius of the convex block. Therefore, the parameter value range of the points on the convex block inclined plane should be limited to:

[0079] Based on the above modeling, the contact point position is effectively determined: First is the contact position between the roller and the lead screw; The meshing of the roller groove surface and the thread surface is as Figure 3 shown. The unit normal vectors of the roller and the lead screw at the contact point are collinear and in opposite directions, that is Then the contact point satisfies the following equation:

[0080] Wherein, r H = r S0 + r R0 represents the revolution radius of the roller,

[0081] and respectively represent the contact deflection angles of the roller and the lead screw, and respectively represent the contact radii of the roller and the lead screw.

[0082] Then comes the contact position between the roller and the nut; the unit normal vectors at the contact point between the roller and the nut are collinear and in opposite directions, that is Then the contact point satisfies the following equation:

[0083] Wherein, and respectively represent the contact deflection angles of the roller and the nut, and respectively represent the contact radii of the roller and the nut.

[0084] Next comes the contact position between the roller and the convex block; the contact between the grooved surface at the bottom of the roller and the inclined surface of the convex block occurs after the roller enters the threadless area of the nut, at which time the meshing between the roller and the lead screw disengages. The angle between the x P axis of the cam ring and the x R axis of the roller is θ P , which is used to simulate the revolution angle of the roller, as Figure 4 shown. The unit normal vector of the convex block at the contact point is:

[0085] The unit normal vectors of the roller and the convex block at the contact point satisfy The meshing equation between the two is:

[0086] Wherein, and respectively represent the contact angle and the contact radius of the roller, and respectively represent the contact angle and the contact radius of the cam ring, is the actual distance between the center of the roller and the axis of the cam ring, that is, the actual revolution radius.

[0087] After the contact position is determined, contact mechanics will also be generated; at the specified contact point, the static force analysis of the single-tooth contact pairs on both sides of the roller is as Figure 5 shown. Since there is no helix angle on the surface of the roller, the normal force is decomposed into axial and radial components, and the expressions are:

[0088] Among them, Q RS and F RSa respectively represent the normal force and axial force on the interface between the roller and the lead screw, while Q RN and F RNa correspond to the normal force and axial force on the interface between the roller and the nut.

[0089] When the roller disengages from the thread meshing with the lead screw and the nut, under the drive of the circular motion of the cage, the roller contacts the inclined surface of the convex block. As Figure 6 shown, the normal force Q RP at the contact point at this time can be represented by the radial force F RPT as: Q RP = F RPT cscβ R ; when the contact point bears the load, its contact area will undergo elastic deformation to form a contact ellipse. According to differential geometry and Hertz contact theory, the major semi-axis and minor semi-axis of the contact ellipse are respectively denoted as a R* and b R* , where *=N, R, P respectively represent the contact between the roller and the nut, the lead screw and the convex block. Therefore, the general expression of the maximum Hertz contact stress σ R* inside the contact ellipse is:

[0090] In the formula, Q R* represents the normal force, while is the reciprocal of the equivalent elastic modulus. and

[0091] are functions related to the contact point, including parameters such as the principal curvature and the eccentricity of the ellipse, and these factors are mainly determined by the structural characteristics of the two contacting bodies.

[0092] After the above structural definitions are completed, a multi-objective optimization model is constructed based on the mathematical model of multi-body contact, and the design structure parameters, optimization objectives, and constraint conditions are determined; a Bayesian learning model is constructed based on variational inference combined with the active learning Kriging strategy;

[0093] Since the contact kinematics of RPRSM involves complex calculations of high dimensions, multi-objectives, and non-linearity, it is necessary to improve the existing Bayesian learning model to solve the problem of high computational cost of traditional Bayesian optimization in high-dimensional problems. Specifically, the VI-ALK strategy can efficiently construct the Bayesian learning model of RPRSM. Referring to Figure 7 shown, it is the flow chart of the VI-ALK strategy. The specific steps are as follows: Step 1: Use the Latin hypercube sampling (LHS) method to perform preliminary sampling in a small-scale sample space, and the obtained initial sampling points are used to construct a surrogate model, that is, to establish a rough Kriging model.

[0094] Step 2: Use the initial Kriging model as the initial estimate of the posterior distribution p(f|D), where f represents the objective function or system response, and D represents the observed data.

[0095] Step 3: Select a new variational distribution q(f; λ) to approximate the true posterior distribution p(f|D), where λ is the variational parameter.

[0096] Step 4: Calculate the evidence lower bound, with the formula as follows:

[0097]

[0098] where, E q(f；λ) denotes the expectation operation with respect to q(f; λ);

[0099] Step 5: Optimize the variational parameter λ * , and find the optimal variational parameter by maximizing the evidence lower bound:

[0100] λ * = argmax L(λ)

[0101] Step 6: Use the optimized posterior distribution q(f; λ * ) to generate a large number of samples and calculate the acquisition function. Specifically, the expected improvement (EI) criterion is adopted, with the formula as follows:

[0102]

[0103] where, x * is the candidate sample obtained by variational inference (VI) sampling, f(x * ) is the predicted objective function value at x * , and f min is the current optimal objective function value. If f(x * ) follows a Gaussian distribution, EI can be expressed as:

[0104]

[0105] where, μ(x * ) and σ(x * ) are the predicted mean and standard deviation of the Kriging model for x * respectively, and φ(·) and Φ(·) represent the probability density function and cumulative distribution function of the standard normal distribution respectively.

[0106] Step 7: Use the Density-based spatial clustering of applications with noise (DBSCAN) method to divide the candidate samples into n different parallel active regions (ARs) in order to select the optimal samples within these regions.

[0107] Step 8: Generate parallel training points in the n active regions These training points are added as candidate points to the optimization process. Select the candidate point with the maximum expected improvement within each active region as the new sampling point for the next iteration. The new sampling point is denoted as x k+1 , and the formula is as follows:

[0108]

[0109] where AR is the active region and EI(x) is the expected improvement function; the set of new sampling points is denoted as The above sampling points are the initial data points used to construct the surrogate model. Training points are the data points used to train the surrogate model, including the initial sampling points and the high-information points added subsequently. Candidate points are the potential optimization points selected during the optimization process and may be added to the training point set. The three are interrelated through the iterative optimization process and jointly promote the improvement of the surrogate model and the realization of the optimization goal.

[0110] Step 9: Check the ratio between the maximum expected improvement (EI) of all samples in the current iteration and the minimum value among the current predicted values to see if it meets the given convergence threshold ε.

[0111]

[0112] where is the minimum objective function value of the current prediction, max(EI) is the maximum expected improvement value of all sampling points, and ε is the given convergence threshold. If it converges, end the process; otherwise, add as an additional training point to the training point set , update the adaptive region and update the ALK model, and then return to Step 2. The new region will gradually shrink to form a circular region centered on the optimal point of the current prediction model. The radius R of the k-th iteration k is as follows:

[0113]

[0114] In the formula, Φ -1 (0.999) represents the value of the inverse cumulative distribution function of the standard normal distribution at 0.999.

[0115] To evaluate the performance of the Bayesian learning model based on the VI-ALK strategy, the Ackley function and the Goldstein-Price function were used for testing and compared with other methods, including traditional Bayesian optimization (BO), particle swarm optimization (PSO), and adaptive simulated annealing (ASA). The optimization process is as Figure 8 shown.

[0116] The Ackley function has a global minimum of f(0,0) = 0, and its mathematical expression is:

[0117]

[0118] The Goldstein-Price function achieves a global minimum at f(0, -1) = 3, and its polynomial expression is:

[0119] f(x,y) = [1 + (x + y + 1) 2 (19 - 14x + 3x 2 - 14y + 6xy + 3y 2 )]·[30 + (2x - 3y) 2 (18 - 32x + 12x 2 + 48y - 36xy + 27y 2 )]

[0120] Both the Ackley function and the Goldstein-Price function have multiple local minima, which may cause the optimization algorithm to fall into local optimal solutions. Within the commonly used domain of these two functions, five initial points generated by Latin hypercube sampling (LSH) are evenly distributed, as Figure 8 (a) and 8(c) shown. The DBSCAN clustering algorithm is applied, restricting the number of clusters to no more than five. The high-information points selected by the acquisition function are used as training points and iteratively added to the model, gradually converging to the optimal value. The convergence threshold is set to 10 -4 . As the adaptive region continues to shrink, the optimization range gradually narrows, and the newly added points gradually concentrate. Finally, the predicted optimal solution approaches the true optimal value, thus verifying the accuracy of the developed model.

[0121] In the multi-objective optimization model constructed above, for the cycloidal planetary roller screw, this application proposes and further optimizes the structural model to determine a set of optimal parameter combinations to minimize the contact stresses between the rollers and the screw, between the rollers and the nut, and between the rollers and the bumps, as follows:

[0122] find x = [k E ,k βN ,k βR ,k βS T ​

[0123] minσ(x,μ) = [σ RN ,σ RS ,σ RP T

[0124] In the formula, the reference roller tooth profile radius scaling factor k E 's definition further introduces the flank angle scaling factors of the nut, roller and lead screw, denoted as k βN , k βR and k βS respectively. These scaling factors are selected as design variables, and their vector form is expressed as: x = [k E , k βN , k βR , k βS T . In addition, other parameters involved in this model are design constants, denoted as μ, and the final objective function can be expressed as: σ(x,μ) = [σ RN , σ RS , σ RP T .

[0125] In the first type of constraint conditions of the multi-objective optimization model, x l and x u represent the lower and upper limits of the design variables respectively. In this example, their value ranges are set to 0.6 to 1.3. In addition, for the tooth profile characteristics, the following three constraint conditions are introduced to regulate the tooth profile to ensure: ① Avoid the tooth tip being too sharp to prevent local stress concentration; ② Prevent the tooth root overlap to ensure a reasonable profile distribution; ③ Eliminate the interference between contact surfaces to ensure smooth meshing. The relevant constraint conditions are as Figure 9 shown. The tooth tip thickness and tooth root thickness of the nut, roller and lead screw are denoted as h ai and h ci respectively, where i = N, R, S corresponds to the nut (N), roller (R) and lead screw (S). Their specific expressions are as follows: r S1 and r S2 are the outer diameter radius and inner diameter radius of the lead screw respectively, r R1 and r R2 are the outer diameter radius and inner diameter radius of the roller respectively, r N1 and r N2 are the outer diameter radius and inner diameter radius of the nut respectively, Ψ N (·), Ψ R (·), Ψ S ​​​(·) are the parametric equations of the tooth profile contours of the lead screw, roller, and nut respectively; the axial clearances between the roller and the corresponding contact surfaces of the lead screw and nut are denoted as and Their mathematical expressions are as follows:

[0126]

[0127] The meshing of the roller with the lead screw (or nut) occurs within the overlapping region of the thread surface and the roller groove surface. The half-angles of the overlapping regions of the roller with the lead screw are denoted as γ RS and γ S , and their expressions are as follows:

[0128] Similarly, the half-angles of the overlapping regions of the roller-nut interface are denoted as γ RN and γ N , and the expressions are as follows:

[0129] This optimization model innovatively introduces additional constraint conditions to limit the contact points on both sides of the roller within a predefined region, corresponding to the last four terms of the optimization model. Specifically: ① The contact angle is limited within the range of the half-angle of the overlapping region of the contact surface; ② The contact radius is restricted within a given range represented by the bandwidth. In this numerical example: The bandwidth of the roller-lead screw interface is denoted as Δ RS , and the bandwidth of the roller-nut interface is denoted as Δ RN , and both are set to 0.1 mm.

[0130] The Bayesian learning model constructed in this application, as a surrogate model for each objective function in the multi-objective optimization model, can efficiently balance the trade-off relationships between multiple objective functions and improve the convergence performance. It is combined with the NSGA-III algorithm. In each iteration, NSGA-III uses these surrogate models to evaluate the objective functions, and finally obtains the optimal parameter combination when the contact points between the structures of the cycloidal planetary roller screw are constrained within a given range and the contact stress is minimized simultaneously.

[0131] The data in Table 1 are used for example analysis. The lead screw and nut are single-start threads. In the multi-body contact model involving the lead screw, roller, and nut, an axial load of 200 N is applied; at the same time, in the contact model between the roller and the bump, a radial force of 120 N is applied to simulate the working conditions of the cycloidal planetary roller screw.

[0132] Table 1 Structural and material parameters of RPRSM

[0133]

[0134] Compared with directly using NSGA-III to solve multi-objective optimization problems, the method of using a surrogate model by NSGA-III to evaluate the objective function significantly reduces the computational time and improves the optimization efficiency. By adopting a fitness assignment mechanism based on hyperplanes to guide the optimization process to stably converge to the Pareto optimal solution set, the optimization results are listed in Table 2. Specifically: the roller profile radius scaling factor k E is increased to 1.0998; the flank angle scaling factors k βN , k βR and k βS of the nut, roller and screw are all reduced by more than 39%. The corresponding optimized flank angles are respectively: the flank angle β N of the nut = 18.24°, the flank angle β R of the roller = 18.04°, and the flank angle β N of the screw = 18.02°. In addition, the values of the optimized objective functions are significantly reduced: the roller-nut contact stress σ RN is reduced to 1416.74 MPa; the roller-screw contact stress σ RS is reduced to 1621.72 MPa; the roller-bump contact stress σ RP is reduced to 1915.82 MPa. These optimization results indicate that the contact performance of the RPRSM has been significantly improved.

[0135] Table 2 Results of multi-objective structural optimization design of cyclic planetary roller screw

[0136]

[0137]

[0138] Figure 10 shows the optimization history of the design variables and the corresponding 3D scatter plot. In Figure 10 (a), the design variables are represented by a colored line graph, the feasible points are marked, and the infeasible points are not marked. The green asterisk represents the optimal point. The iterative process shows fluctuations at the beginning and finally converges to the optimal solution. Figure 10 (b)–(d) show the distribution of 241 iterative points on the fitting surfaces of k E , β R and the objective function. The black dots represent the feasible points and are relatively concentrated; the red dots represent the infeasible points and are relatively scattered. Although the objective function is affected by other design variables, the good distribution of the scatter points on the fitting surface indicates that k E and β R are the main factors affecting the contact stress. In addition, Figure 10 (b) has the steepest σ RN gradient on the fitting surface, indicating that the fluctuation range of the contact stress between the roller and the nut is relatively large, while Figure 10The fitting surface in (d) is the flattest, reflecting that the contact stress fluctuation range between the roller and the bump is the smallest.

[0139] Figure 11 The objective function optimization history curve and the distribution of iteration points in the objective function space are displayed. Figure 11 The curves in (a) show a similar pattern to the design variables. The distribution of the scattered points in the space composed of σRN, σRS, and σRP also shows good convergence, and the optimal point is beneficial to all three objective functions.

[0140] During the optimization process, the most frequent penalty comes from violating the contact position constraint, followed by interference between the two surfaces at the contact point caused by the design variables. Therefore, ensuring the optimal performance of the contact point is crucial for the structural design of the recirculating planetary roller screw. Figures 12 - 14 The comparison of the local contact characteristics between the roller and the screw, nut and bump after optimization is presented, indicating that both the semi-major axis and the semi-minor axis of the contact ellipse are expanded.

[0141] Figure 12 It shows that the contact point between the roller and the screw moves down along the y-axis by about 0.04 mm and remains on the pitch circle in the x-direction. The contact point between the roller and the nut moves to the right by 0.045 mm and upward by 0.04 mm compared to the initial position, and both are within the constraints of the optimization model, such as Figure 13 The optimized parameters reduce both the normal force and the sum of the principal curvatures, thus promoting the optimization effect. The maximum contact stress in the contact ellipse of the roller-screw and roller-nut is finally reduced by 31.94% and 32.49%, respectively.

[0142] The contact position between the roller and the bump changes little, with only a slight downward shift in the y direction, e.g. Figure 14 As shown. In addition, the increase in the contact ellipse area between the roller and the bump is the largest among the three pairs of contacts, although the reduction in contact stress is the smallest, at 17.36%. This is because the model applies radial force instead of axial force to simulate the rotational motion of the roller under the thrust of the cage. The optimized parameters reduce the sum of the principal curvatures but increase the normal force, thus achieving a better optimization balance in contact stress.

[0143] The above embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Any equivalent substitution or change made by a person skilled in the art based on the present invention is within the protection scope of the present invention.

Claims

1. An optimized design method for a cyclic planetary roller screw structure, characterized in that, It includes the following steps: Establish a mathematical model of multi-body contact for a cyclic planetary roller screw; Construct a multi-objective optimization model based on the mathematical model of multi-body contact, and determine the design structure parameters, optimization objectives, and constraint conditions; Construct a Bayesian learning model based on variational inference combined with an active learning Kriging strategy; By combining the Bayesian learning model with the NSGA-III algorithm, solve the multi-objective optimization model to obtain the optimal parameter combination when the contact points between the structures of the cyclic planetary roller screw are constrained within a given range and the contact stress is minimized simultaneously.

2. The optimized design method for the cyclic planetary roller screw structure according to claim 1, characterized in that The external thread profile of the screw of the cyclic planetary roller screw is triangular. The internal thread profile of the nut of the cyclic planetary roller screw is the same as that of the screw and has a non-threaded area. The rollers of the cyclic planetary roller screw are non-threaded structures and are equidistant groove-shaped structures, and the groove pitch of the rollers is equal to the lead of the screw. A pair of cam rings of the cyclic planetary roller screw are installed at both ends of the nut and coincide with the axis of the nut. There are bumps on the cam rings, the height of the bumps is equal to the lead of the screw, and the inclination angle of the bumps is equal to the flank angle of the rollers.

3. The optimized design method of the cyclic planetary roller screw structure according to claim 2, characterized in that, The mathematical model of multi-body contact includes: The parametric equation of the tooth profile of the screw to determine the unit normal vector of any point on the screw thread tooth; The parametric equation of the tooth profile of the nut to determine the unit normal vector of any point on the nut thread tooth; The parametric equation of the tooth profile of the roller to determine the unit normal vector of any point on the roller groove surface; The parametric equation of the inclined surface of the bump to determine the parameter value range of the points on the inclined surface of the bump.

4. The optimized design method for the cyclic planetary roller screw structure according to claim 3, characterized in that, Determine the contact positions and contact stresses between the rollers and the screw, between the rollers and the nut, and between the rollers and the bumps.

5. The optimized design method for the cyclic planetary roller screw structure according to claim 1, characterized in that The design structure parameters include the scaling factor of the roller tooth profile radius, the flank angle of the roller, the flank angle of the screw, and the flank angle of the nut, where the inclination angle of the bump on the cam ring is equal to the flank angle of the roller; The optimization objectives include the contact stress between the roller and the screw, the contact stress between the roller and the nut, and the contact stress between the roller and the bump; The constraint objectives include tooth profile constraint, axial clearance constraint, and contact point constraint. The tooth profile constraint is used to limit the tooth top thickness and tooth root thickness. The axial clearance constraint is used to prevent interference between the rollers and the screw and between the rollers and the nut. The contact point constraint is used to accurately limit the contact point position.

6. The optimized design method for the cyclic planetary roller screw structure according to claim 1, characterized in that The multi-objective optimization model after setting the constraint conditions is: find x=[k E ,k βN ,k βR ,k βS T ​ minσ(x,μ)=[σ RN ,σ RS ,σ RP T ​ where k E is the roller tooth profile radius scaling factor, k βN , k βR and k βS are the flank angle scaling factors of the nut, roller and lead screw respectively, x is a vector, μ is a design constant, and σ(x, μ) is the final objective function; h cN 、h cR 、h cS represent the crest thicknesses of the nut, roller, and lead screw respectively, and h aN 、h aR 、h aS represent the root thicknesses of the nut, roller, and lead screw respectively; P N 、P R 、P S represent the pitches of the nut, roller, and lead screw respectively; and represent the axial clearances between the roller and the corresponding contact surfaces of the lead screw and nut respectively, and γ RS and γ S represent the half-angles of the overlapping regions between the roller groove surface and the lead screw thread surface respectively; γ RN and γ N represent the half-angles of the overlapping regions between the roller groove surface and the nut thread surface respectively; r S0 is the nominal radius of the lead screw, r N0 is the nominal radius of the nut, and Δ RS is the bandwidth of the fluctuation range of the contact point position between the roller and the lead screw, and Δ RN is the bandwidth of the fluctuation range of the contact point position between the roller and the nut; and represent the contact deflection angles of the roller and the nut respectively, and represent the contact radii of the roller and the nut respectively; and represent the contact deflection angles of the roller and the lead screw respectively, and represent the contact radii of the roller and the lead screw respectively.

7. The optimized design method of the cyclic planetary roller screw structure according to claim 6, characterized in that The tooth top thickness and tooth root thickness meet the following requirements: r S1 and r S2 are the outer diameter radius and inner diameter radius of the lead screw respectively, r R1 and r R2 are the outer diameter radius and inner diameter radius of the roller respectively, r N1 and r N2 are the outer diameter radius and inner diameter radius of the nut respectively, Ψ N (·), Ψ R (·), Ψ S (·) are the parametric equations of the tooth profile contours of the lead screw, roller and nut respectively. The axial clearance between the roller and the corresponding contact surfaces of the screw and the nut meets the following requirements: The half angle of the overlapping area between the column and the screw meets the following requirements: The half angle of the overlapping area of the roller and nut interface meets the following requirements:

8. The optimal design method for the cyclic planetary roller screw structure according to claim 1, characterized in that The sampling strategy of variational inference combined with active learning Kriging is as follows: Step 1: Use the Latin hypercube sampling method to conduct preliminary sampling in a small-scale sample space to establish a rough initial Kriging model; Step 2: Use the initial Kriging model as the initial estimate of the posterior distribution p(f|D), where f represents the objective function and D represents the observed data; Step 3: Select a new variational distribution q(f; λ) to approximate the true posterior distribution p(f|D), where λ is the variational parameter; Step 4: Calculate the evidence lower bound; Step 5: Find the optimal variational parameter λ by maximizing the evidence lower bound * ; Step 6: Use the optimized posterior distribution q(f; λ * ) to generate a large number of candidate samples; Step 7: Use the density-based spatial clustering method to divide the candidate samples into n different and parallel active regions; Step 8: Generate parallel training points in n active regions These training points are added to the optimization process as candidate points, and the candidate point with the maximum expected improvement is selected as the new sampling point for the next iteration within each active region; Step 9: Check the ratio between the maximum expected improvement of all samples in the current iteration and the minimum value among the current predicted values to determine whether the given convergence threshold is met. If it is met, end; if not, add the new sampling point as an additional training point to the training point set, update the adaptive region and update the Kriging model, and then return to Step 2 to continue execution.

9. The optimized design method for the cyclic planetary roller screw structure according to claim 1, characterized in that Take the Bayesian learning model as the surrogate model for the optimization objective in the multi-objective optimization model, and the NSGA-III algorithm uses the surrogate model to evaluate the objective function.

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